from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(101, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([37]))
pari: [g,chi] = znchar(Mod(96,101))
Basic properties
Modulus: | \(101\) | |
Conductor: | \(101\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 101.h
\(\chi_{101}(4,\cdot)\) \(\chi_{101}(9,\cdot)\) \(\chi_{101}(13,\cdot)\) \(\chi_{101}(20,\cdot)\) \(\chi_{101}(21,\cdot)\) \(\chi_{101}(22,\cdot)\) \(\chi_{101}(23,\cdot)\) \(\chi_{101}(30,\cdot)\) \(\chi_{101}(33,\cdot)\) \(\chi_{101}(43,\cdot)\) \(\chi_{101}(45,\cdot)\) \(\chi_{101}(47,\cdot)\) \(\chi_{101}(49,\cdot)\) \(\chi_{101}(64,\cdot)\) \(\chi_{101}(70,\cdot)\) \(\chi_{101}(76,\cdot)\) \(\chi_{101}(77,\cdot)\) \(\chi_{101}(82,\cdot)\) \(\chi_{101}(85,\cdot)\) \(\chi_{101}(96,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{25})\) |
Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\(2\) → \(e\left(\frac{37}{50}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 101 }(96, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(-1\) | \(e\left(\frac{31}{50}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)